Linear Spaces
Linear (Vector) Spaces
Definition: A linear space (or vector space) over a field F is a set V with two operations, addition and scalar multiplication such that the following axioms hold:
Vector Addition (+):V×V→V such that x+y∈V for all x,y∈V.
- (A1) x+y=y+x ∀x,y∈V (commutativity)
- (A2) x+(y+z)=(x+y)+z ∀x,y,z∈V (associativity)
- (A3) There exists an element 0∈V such that x+0=x ∀x∈V (existence of additive identity)
- (A4) For every x∈V there exist an element −x∈V such that x+(−x)=0 ∀x∈V (existence of additive inverse)
Scalar Multiplication (⋅):F×V→V such that αx∈V for all α∈F and x∈V.
- (M1) α(bx)=(αb)x ∀α,β∈F and ∀x∈V (associativity)
- (M2) α(x+y)=αx+αy ∀α∈F and ∀x,y∈V (distributivity)
- (M3) (α+β)x=αx+βx ∀α,β∈F and ∀x∈V (distributivity)
- (M4) 1F​x=x ∀x∈V (existence of multiplicative identity)
Elements of the vector space are called vectors, or points.
Scalar multiplication in a vector space depends on the field F. Thus when we say x∈V, we mean that x is a vector in the vector space V over the field F or (V,F).
For example, Rn is a vector space over the field R, and Cn is a vector space over the field C.
Sharing the same field is a necessary condition for two vector spaces to be comparable. For example, Rn and Cn are not comparable.
The simplest vector space contains only one point. In other words, {0} is a vector space.
Example: Show that x0F​=0V​ for all x∈V
Proof:
- y=x0F​=0V​ is assumed to be true.
- x+y=x+x0F​=x1F​+x0F​ (M4)
- x+y=x(1F​+0F​)=x1F​ (M3)
- x+y+(−x)=x1F​+(−x1F​) (A4)
- y=x(1F​+(−1F​))=x0V​=0V​ (A2) â–
Example: Show that x0F​=0v​
Proof:
- Assume y=x0F​ and show y=0V​.
- x+y=x+x0F​=x1F​+x0F​ (M4)
- x+y=x1F​+x0F​=x(1F​+0F​) (M3)
- x+y=x(1F​+0F​)=x1F​ (A3)
- x+y=x1F​=x (M4)
- x+y+(−x)=x+(−x) (A4)
- y=x+(−x)=0V​ (A4) â–
Remark: A vector space has a unique additive identity.
Function Space
Definition: Let S be a set and F be a field. The set of all functions from S to F is denoted by FS.
For f,g∈FS, the sum f+g∈FS is the function defined by
(f+g)(x)=f(x)+g(x), ∀x∈S
For f∈FS and α∈F, the product αf∈FS is the function defined by
(αf)(x)=αf(x), ∀x∈S
Example: Set of all polynomials with degree n with coefficients in F is denoted by F[x]n​.
What essentially function spaces are, is that they are the set of all functions from a set to a field. For example, the set of all polynomials with degree n with coefficients in F is denoted by F[x]n​.
#EE501 - Linear Systems Theory at METU